What Shapes Do You See? Student Explorations in Mathematics What Shapes Do You See? What Are Figurate Numbers? While the rest of humanity was seemingly occupied with the more practical uses of mathematics, the Greeks were among the first to develop a mathematical world that was not necessarily tied to real-world applications, including what were to become figurate numbers. A. Consider the three arrangements in 2. How many dots would be in arrangements 4 and 5? figure .1 3. Identify the shapes you notice in the arrangements. Figure 1 4. Write a description of how these arrangements 1. Predict the next two arrangements in this growing are constructed. Share your answers with a pattern. partner. We acknowledge Kathy Clark, Florida State University, for her submission of the idea for this issue. Copyright © 2012 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only. What Shapes Do You See? 5. Invent a name for these figurate numbers. 6. Consider the shapes you found in question B3. Sketch another way to group these dots into geometric shapes. 6. Consider the shapes you found in question A3. Sketch another way to group these dots into geometric shapes. C. The shapes in figure 3 are arrangements 4 and 5 in the sequence of a growing pattern. B. Consider the three arrangements in figure .2 Figure 3 1. What are the first three arrangements in this growing pattern? (Remember that figure 3 shows arrange- ments 4 and 5.) Figure 2 1. Predict the next two arrangements in this growing pattern. 2. How many dots would be in arrangements 1, 2, and 3? 2. How many dots would be in arrangements 4 and 5? 3. Identify the shapes you notice in the arrangements. 3. Identify the shapes you notice in the arrangements. 4. Explain what you observe about how these arrangements are constructed. Share your 4. Explain what you observe about how these answers with a partner. arrangements are constructed. Share your answers with a partner. 5. Invent a name for these figurate numbers. 5. Invent a name for these figurate numbers. 2 Student Explorations in Mathematics, January 2012 What Shapes Do You See? 6. Consider the shapes you found in question C3. Sketch 3. Sketch the shapes you notice in the arrangements. another way to group these dots into geometric shapes. 4. Describe how these arrangements are constructed. The patterns in figures 1–3 reveal what mathematicians call figurate numbers. Figurate numbers get their name from the fact that they model geometric shapes or figures. Now, make up your own arrangement of some figurate numbers and share with another student in your classroom. 5. You may see two or more shapes within the overall shapes. Sketch the shapes inside of other shapes and share your drawings with a partner. D. Consider the figurate number sequence in figure 4. 6. Use your observations to predict how many dots would be in arrangement 10 of this sequence. 7. Tell a partner how you determined your answer to question D6. Figure 4 1. What are the next two arrangements in this growing pattern? 8. Invent a name for these figurate numbers. 9. Consider the shapes you found in question D3. Sketch another way to group these dots into geometric shapes. 2. How many dots would be in arrangements 6 and 7? 3 Student Explorations in Mathematics, January 2012 What Shapes Do You See? E. The figurate number sequence in figure 5 3. Describe how these arrangements are constructed. contains arrangements 4 and 5. 4. You may see two or more shapes within the overall shapes. Sketch the shapes inside of other shapes that you notice in this sequence. Figure 5 1. Sketch the first three arrangements in this growing pattern. 5. Use your observations to predict how many dots would be in arrangement 20 of this sequence. 6. Tell a partner how you determined your answer to question E5. 7. Invent a name for these figurate numbers. 2. Sketch what you think arrangements 6 and 7 would look like if this were a growing pattern. 8. Consider the shapes you found in question E4. Sketch another way to group these dots into geometric shapes. F. Consider all the sequences in figure ,6 which is actually figures 1–5combined. 1. How are the sequences the same? 4 Student Explorations in Mathematics, January 2012 What Shapes Do You See? Figure 6 (Combined Figures 1–5) 2. How are the sequences different? 2. Sketch how the dots can be arranged. What shapes or figures emerge? Sketch what the fifth arrangement might look like. Share your sketches with a partner. 3. Explain what you observe about how these arrange- ments are constructed. H. Consider the sequence in figure .7 4. Which sequences are embedded in other sequences? Share your answers with a partner. G. Suppose that all you knew was the number of dots in a sequence of arrangements in a growing pattern. Figure 7 1. If the number of dots in the first four arrangements were 1. How is figure 7 the same as or different from your 1, 5, 13, and 25, respectively, sketch how you think the sketches in part G? dots might be arranged and what shapes might be possible. 5 Student Explorations in Mathematics, January 2012 What Shapes Do You See? Mathematicians named the sequence in figure 7 centered J. On the basis of what you have explored and square numbers. learned in this activity, complete table 1 (see p. 7). I. Using sketch paper or an electronic device, 1. Share your numerical and verbal representa- tions of figurate number tables with other create your own arrangements to illustrate a students, in pairs or in groups of three. figurate number sequence. Figure 8 shows an example of centered square numbers created using Doodle Buddy on an iPad®. 2. Explain your understanding of patterns created by another student in your group. Figure 8 1. Explain the mathematics that describes your sequence. 3. How are these patterns the same? How are they different? 2. How do your arrangements change from one arrangement to the next? 4. Some types of figurate numbers can be even or odd. Can any type of figurate number never be even? Why, or why not? 3. Describe how you can use what you know about your sequence to predict the 100th arrangement; the 568th arrangement; the 1000th arrangement; and any arrangement. 5. Concerning the values and verbal explanations in table 1, what are you sure about? What are you unsure of? What do you wonder about the values and verbal explanations in table 1? 6 Student Explorations in Mathematics, January 2012 What Shapes Do You See? Table 1. Numerical and Verbal Representations of Figurate Numbers Figurate Square Pentagonal Centered Square Student-Created Number Type Numbers Numbers Numbers Pattern Number of dots 1, 4, 9, 16, 25 1, 5, 12, 22, 35 1, 5, 13, 25, 41 in each arrangement Verbal: Briefly describe how each figurate number is determined. (Note: There may be more than one way to express this, which we will address later in the exploration.) K. Create your own figurate number notation. L. Explore common figurate number notation 1. Think about how you described the number sequence agreed on by mathematicians. in table 1. Invent a way to use notation to represent the T4 means the fourth triangular number: T stands for trian- sequences of square numbers, pentagonal numbers, gular; the subscript 4 represents the fourth such number. and centered square numbers. (As a side note, the subscript is sometimes called the index.) The subscripted notation for representing figurate numbers is useful. It is easier to write T4 than the fourth triangular number. There is another interesting way to represent figurate 2. Using words, describe how your notation works. numbers, which is called the symbolic (or algebraic) rep- resentation. We can represent the nth triangular (or any figurate number) by determining the algebraic formula for it. For example, we can examine how to calculate the nth tri angular number Tn, where n is a natural number, by using analysis: T8 = the sum of the first eight natural numbers, or 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36. Complete table 2 to reflect the total value of each tri angular number. 3. Illustrate samples of how your notation works. Consult with others in your group and find 1. Explain how the numbers in table 2 are determined. other notations that are similar to or different from your notation. Table 2. Triangular Numbers Triangular no. index n 1 2 3 4 5 … 21 22 23 24 Triangular no. Tn 1 3 7 Student Explorations in Mathematics, January 2012 What Shapes Do You See? 2. Examine table 2 for a pattern in determining the value of each triangular number. Determine the value of the fiftieth triangular number using your pattern. ➧ 3. Write an equation that would predict the number of dots for any triangular number. Figure 11 N. Consider these points. 1. How is the Handshake problem similar to or different from what you have seen about figurate numbers? M. Explore a representation of the Handshake problem. If everyone in the room shook hands with everyone else, how many handshakes would there be? 2. Predict how many handshakes there would be in a Suppose we represent triangular number 4, or T , with lady- 4 room with 50 people, 100 people, 1000 people, and bugs rearranged to form a staircase-like figure (seefig. 9). any number of people. O. Let’s make a few conjectures. Figure 9 Let’s examine some conjectures about relationships among the different figurate numbers.